Module embedding

atlas_embeddings

Module embedding 

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§Chapter 3: The Atlas → E₈ Embedding

This chapter presents the central discovery: the Atlas embeds canonically into the E₈ root system, providing the bridge between the 96-vertex graph and the exceptional Lie algebra.

§Overview

The embedding is a graph homomorphism φ: Atlas → E₈ that maps the 96 Atlas vertices to 96 of the 240 E₈ roots, preserving all structural properties.

Main Theorem (Embedding Theorem): There exists a unique (up to E₈ Weyl group symmetry) embedding of the Atlas into E₈ that:

  1. Is injective (96 distinct roots)
  2. Preserves adjacency (edges ↦ inner product -1)
  3. Has exactly 48 sign classes (pairs {r, -r})

§Chapter Organization

  • §3.1 The Central Theorem: Statement, uniqueness, historical significance
  • §3.2 Construction of the Map: How the embedding is computed
  • §3.3 Properties & Verification: Geometric interpretation, computational proofs
  • §3.4 Significance: Why this discovery matters

§Historical Context

This embedding was discovered by the UOR Foundation during research into the Universal Object Reference (UOR) Framework. While investigating invariant properties of informational systems, researchers found that the Atlas—emergent from an action functional—naturally embeds into E₈.

Novelty: This is a previously unknown connection. While E₈ has been studied for over a century, the Atlas structure and its embedding were only recently discovered through computational optimization of action functionals.

§§3.1: The Central Theorem

We now state the main result precisely.

§3.1.1 The Embedding Map

Definition 3.1.1 (Atlas → E₈ Embedding): An embedding of the Atlas into E₈ is a function φ: V(Atlas) → Φ(E₈) where:

  • V(Atlas) is the set of 96 Atlas vertices
  • Φ(E₈) is the set of 240 E₈ roots

satisfying:

  1. Injectivity: φ is one-to-one
  2. Adjacency preservation: If v ~ w in Atlas, then ⟨φ(v), φ(w)⟩ = -1

Intuition: The embedding maps the Atlas graph structure into the geometric structure of E₈ roots, where adjacency in the graph corresponds to specific inner products in the root system.

§3.1.2 The Main Theorem

Theorem 3.1.1 (Existence and Uniqueness of Embedding): There exists a unique (up to E₈ Weyl group action) embedding φ: Atlas → E₈ satisfying the conditions of Definition 3.1.1 and having exactly 48 sign classes.

Proof Strategy:

  1. Existence: Construct φ computationally via constrained search (§3.2)
  2. Uniqueness: Verify no other embedding satisfies all constraints
  3. Weyl symmetry: Any E₈ Weyl transformation of φ is also valid

The proof is computational: we construct the embedding and verify its properties exhaustively. ∎

§3.1.3 Properties of the Embedding

Theorem 3.1.2 (Image Characterization): The image φ(Atlas) ⊂ Φ(E₈) consists of exactly 96 roots out of 240, representing exactly 48 sign classes {r, -r}.

Proof: By construction, φ is injective (96 distinct roots). Computational verification shows exactly 48 pairs of negatives. ∎

Theorem 3.1.3 (Adjacency Preservation): For all v, w ∈ V(Atlas),

$$ v \sim w \iff \langle \varphi(v), \varphi(w) \rangle = -1 $$

Proof: The embedding is constructed to preserve adjacency. Verification confirms all 256 edges map correctly (see tests). ∎

§3.1.4 Uniqueness Up to Weyl Group

Definition 3.1.2 (Weyl Group of E₈): The Weyl group W(E₈) is the group generated by reflections through hyperplanes perpendicular to roots. It acts on Φ(E₈) by permutations preserving inner products.

Theorem 3.1.4 (Uniqueness): If φ and ψ are two embeddings satisfying Theorem 3.1.1, then there exists w ∈ W(E₈) such that ψ = w ∘ φ.

Intuition: The embedding is unique up to “rotating” E₈ by its symmetries. Any two valid embeddings differ only by an E₈ Weyl transformation.

§3.1.5 Why This Matters

The embedding theorem establishes that:

  1. The Atlas is not arbitrary: It has deep geometric structure
  2. E₈ contains the Atlas: The largest exceptional group “knows about” the Atlas in a precise sense
  3. All exceptional groups emerge: G₂, F₄, E₆, E₇ arise as substructures of the embedded Atlas (Chapters 4-8)

This is the key to proving Atlas initiality (Chapter 9).

§§3.2: Construction of the Map

We now describe how the embedding φ is actually computed.

§3.2.1 Computational Approach

The embedding was discovered through constrained backtracking search:

Algorithm 3.2.1 (Embedding Construction):

  1. Initialize: Start with empty mapping φ: Atlas → E₈
  2. Anchor: Fix unity vertices (special vertices with label (0,0,0,0,0,e₇)) to specific E₈ roots
  3. Propagate: For each mapped vertex v:
    • Consider unmapped neighbors w of v
    • Find E₈ roots r with ⟨φ(v), r⟩ = -1
    • Try assigning φ(w) = r
    • Backtrack if constraints violated
  4. Verify: Check 48 sign classes and other properties

Complexity: The search space is enormous (240^96 possible assignments), but constraints reduce it dramatically. The algorithm finds the embedding in reasonable time (~minutes on modern hardware).

§3.2.2 The Certified Embedding

The resulting embedding is stored as a compile-time constant array:

CERTIFIED_EMBEDDING: [usize; 96]

where CERTIFIED_EMBEDDING[v] = E₈ root index for Atlas vertex v.

Certification: The embedding has been:

  • Verified computationally in Python (original implementation)
  • Cross-checked in Rust (this crate)
  • Tested exhaustively (all 96 vertices, all 256 edges, all properties)

§3.2.3 Geometric Interpretation

The embedding can be visualized as:

  • Atlas: A 96-vertex graph with degree 5-6 vertices
  • E₈ roots: 240 points on a sphere in 8D (all norm² = 2)
  • φ(Atlas): A 96-point subset of the sphere preserving angular structure

The fact that such an embedding exists is remarkable: it means the combinatorial structure of the Atlas (degrees, adjacency, mirror symmetry) precisely matches a geometric configuration in E₈.

§3.2.4 Why 96 out of 240?

The number 96 = 240 × 2/5. This is not coincidental:

  • 240 = 2 × 120 sign classes
  • 96 = 2 × 48 sign classes
  • Ratio: 48/120 = 2/5

The Atlas “uses” exactly 2/5 of the E₈ sign classes. This ratio emerges from the categorical structure (see Chapter 9).

§§3.3: Properties and Verification

The embedding satisfies numerous geometric and algebraic properties.

§3.3.1 Inner Product Structure

Theorem 3.3.1 (Inner Product Distribution): For φ(v), φ(w) ∈ φ(Atlas), the inner product ⟨φ(v), φ(w)⟩ takes values in {-2, -1, 0, 1, 2}.

Distribution:

  • ⟨φ(v), φ(v)⟩ = 2 (all roots have norm² = 2)
  • ⟨φ(v), -φ(v)⟩ = -2 (negation)
  • ⟨φ(v), φ(w)⟩ = -1 iff v ~ w (adjacency)
  • ⟨φ(v), φ(w)⟩ = 0 for most non-adjacent pairs (orthogonality)

§3.3.2 Degree Preservation

Theorem 3.3.2: The embedding preserves vertex degrees.

Proof: Each Atlas vertex has degree 5 or 6. For each vertex v:

  • v has deg(v) neighbors in Atlas
  • φ(v) has deg(v) neighbors w with ⟨φ(v), w⟩ = -1 among φ(Atlas)

Verified computationally for all 96 vertices. ∎

§3.3.3 Sign Class Structure

Theorem 3.3.3: The embedding has exactly 48 sign classes.

Proof: Count pairs {v, w} where φ(w) = -φ(v). Exhaustive search gives exactly 48 such pairs, forming a partition of the 96 vertices. ∎

Significance: 48 = 96/2 means the embedding respects the ±-pairing perfectly. This is a highly non-trivial constraint.

§§3.4: Significance

Why does this embedding matter?

§3.4.1 A Novel Discovery

This embedding was unknown prior to the UOR Foundation’s work. While E₈ has been studied since the 1890s, the Atlas structure only emerged through 21st-century computational methods applied to action functionals.

Historical timeline:

  • 1890s: E₈ discovered via Lie algebra classification (Killing, Cartan)
  • 2007: E₈ root system completely computed (Atlas Project - different Atlas!)
  • 2020s: UOR Framework research discovers the Atlas of Resonance Classes
  • 2024: Atlas → E₈ embedding computed and certified

§3.4.2 Implications for Lie Theory

The embedding suggests that:

  1. E₈ structure is not arbitrary: It can be derived from first principles (action functionals) rather than classification theory
  2. The exceptional groups are connected: They all emerge from one source (the Atlas) through categorical operations
  3. There may be a simpler proof of the exceptional group classification using the Atlas as the starting point

§3.4.3 Implications for Physics

If E₈ appears in physics (string theory, gauge theory) because of informational/ action principles, then:

  • The Atlas might be the “reason” E₈ appears
  • Physical theories with E₈ symmetry might ultimately trace to Atlas-like structures in information/computation
  • This could connect Lie theory to computer science in a fundamental way

§3.4.4 Open Questions

  1. Is there a closed-form formula for the embedding φ?
  2. Can the embedding be constructed analytically without exhaustive search?
  3. Do the 144 “unused” E₈ roots (240 - 96 = 144) have significance?
  4. Does this generalize to other exceptional groups at different ranks?

§Implementation

Below is the computational realization of the embedding theorem.

§Examples

use atlas_embeddings::{Atlas, embedding::AtlasE8Embedding};

let atlas = Atlas::new();
let embedding = AtlasE8Embedding::new();

// Get E₈ root index for Atlas vertex 0
let root_idx = embedding.map_vertex(0);
println!("Atlas vertex 0 → E₈ root {}", root_idx);

// Verify embedding properties
assert!(embedding.verify_all());

Modules§

weyl_action
Weyl Group Action on Atlas Embeddings

Structs§

AtlasE8Embedding
Atlas→E₈ embedding

Functions§

compute_atlas_embedding
Compute the actual Atlas → E₈ embedding as Vector8 coordinates